One may already comment on the origin of the axioms. The first one illustrates the scattering properties of the model on the level of the operator's kernel.
The second and third axioms may be interpreted heuristically as a consequence of the LSZ reduction [1]}, and locality of the operator,
see e.g. [2]}, [3]} for heuristics on that matter. Finally,
the last axiom is a manifestation of the Lorentz invariance of the theory. The number \(  \texttt {s}  _{  \texttt {O}  }\)  arising in \(\mathrm {iv)}\)  is called the spin of the operator.
Moreover, the number \(\ell \)  depends on the type of operator being considered.
Finally, for more complex models, one would also need to add an additional axiom which would encapsulate the way how the presence of bound states
in the model governs certain additional poles in the form factors, c.f. [3]}.
